Saturday, July 1, 2017

Teaching Geometry Proofs


One of the biggest ideas in high school geometry is getting students to write a two column proof.  Although proofs seem to be emphasized less these days, it is still an important concept to get across to students.  Step by step logical argument is an important skill that can be applied to any career or aspect of life.

I like to start teaching proofs with something the students can relate to.  I ask one student to come to the front of the room.  I ask him or her to take of their shoe {must be a tie shoe!}.  I then ask for a volunteer to give that student verbal directions on how to tie his or her shoe.  I tell the demonstrating student that they must do exactly what the directions say.  I use this exchange so students will hopefully see how important it is to be very specific and step-by-step with their directions.

Next, I like to use a couple of puzzles to continue to develop the idea that a step by step process is necessary when writing a proof.  I use word puzzles that are sometimes called word ladders.



We then move on to another thing that students are very familiar with - solving algebra equations.  I setup several algebra equations in two column form.  Students get familiar with the format of a proof and start to understand how to move step by step through the process.




Finally, I feel that my students are ready for the big event...writing an actual geometry proof.  Unfortunately, this sometimes doesn't live up to what I might have built it up to be.

As we move through the properties and apply them to geometric situations instead of algebraic ones, students start to get the idea.

Soon, we make it to congruent triangle proofs where these ideas are really applied.  Introducing the ideas of congruent triangles slowly with MANY examples helps students get the idea.  Using lots of different activities {task cards, board work, fill in the blank, etc}

I just finished making a flip book that I am going to use this year to introduce Congruent Triangle Proofs with CPCTC.  One example on each page that students can fill in as we go.  When we are finished, they can store it in their notebook by gluing down the back page if they wish.


Below are some links to products that you can use to help your students work on writing proofs in geometry.

Good luck!




Sunday, June 25, 2017

Digital Task Cards for Math Class



Digital task cards in math class?

How can I continue to use task cards when we are going to a 1:1 environment?



Here are two ways to enjoy the benefits of task cards digitally:

1) Use your task cards one at a time as entrance or exit tickets.

For example,

Here is a recent task card that I made for my geometry class.



At first glance, this might seem like a regular task card you might hand out to your class.  But, upon closer inspection, notice, that there are places where the students can insert text boxes to write an equation, find the value of x, and find the measure of a specified angle.

You can assign the students this one task card at the beginning or end of a class period.  They send their answer back to you electronically and you immediate see how everyone is doing!  No papers to flip through (or accidentally misplace!) 

2) You can use your task cards with Google Forms.  Here is an example of a Google form with a task card inserted.


Students see the task card image as part of their question and then choose their answer below the card.  You can make multiple choice questions or students can type their answer.  This is even better than sending the students task cards one at a time because you get immediate data from Google!  A great way to formatively assess students!

Just like anything else, task cards in a 1:1 environment have their place.  Having the students write in mathematical notation is still a major stumbling block.  But for now, I hope you can find a place to use these two options in your class.

{ Want to see more digital activities in my store?  Check out this link: Digital Math Activities }

Sunday, April 30, 2017

AP Saturday - An All School Workshop



I had been thinking all winter about how to entice my AP Calculus students to come in on a {gasp} Saturday or Sunday to review with me.  I know they are all busy...they are in many activities and have many other AP classes to study for.

So, I brainstormed with the other AP teachers at my school, and we came up with the idea of AP Saturday.  Here's the general idea:

We started the day off in our Leadership Center with a general pep talk to all of the students.


After the pep talk, students were dismissed to attend half hour sessions of their own choosing.  Each teacher that was present divided the two and a half time period into 5 specific sessions.  Students were given a schedule that listed what each teacher was planning for each half hour session.  (For example - from 9:30 - 10:00 my students and I worked on reviewing area and volume - from 10:00 to 10:30 we reviewed separable differential equations and so on.)  Publishing ahead of time what we each planned on talking about allowed students to pick and choose what they felt they needed to attend.

I bribed gave my students 2 points extra credit for each session they attended.  Some students came, some didn't, but I believe this offered students the opportunity to review and ask questions in a relaxed environment.

Involving other AP Teachers in the school got more people involved and enticed more students to come and study :)

We ended the day by having pizza together in the cafeteria.




Sunday, March 5, 2017

Pi Day Activities for Secondary Math


I LOVE PI DAY!

Who doesn't love a chance to celebrate in the math classroom?

Here are links to a bunch of activities that could be used in your Secondary Math Classroom.



Saturday, February 18, 2017

Catalog of Desmos Activities For High School Math



This is a post where I will keep a catalog of activities I have created to use with Desmos.  There is a link up at the bottom of the post where you can get a direct link to the activity, but I would like to describe the activities here.

1)  Pythagorean Theorem Triangle Family Sort - In this activity, students are given the headings 3, 4, 5; 5, 12, 13; 7, 24, 25; and 8, 15, 17.  Students sort the cards into appropriate piles based on the family that the triangle belongs to.

2)  Parabola Card Sort - In this activity, students are given graphs of parabolas.  They match the graphs to the equation of the parabola, the focus and the directrix.

3) Law of Sines - Ambiguous Case - Students decide whether the information given will allow 0, 1, or 2 triangles to be formed.

If you happen to find any of these activities useful, you are welcome to use them in your classroom too :)





Saturday, January 28, 2017

Proving Math Formulas to High School Students


I am teaching PreCalc this year.  We are knee deep in our trig unit and I am planning to teach Trig Angle Addition formulas on Monday.  I seriously dread this day because I don't see any motivation for it.  I mean we have a calculator to figure out what sin 75˚ or cos 165˚ is...who cares about the exact value???

So, I checked around the internet to see if I could find some motivation and I seriously still haven't found too many reasons why we are still actually teaching this topic, BUT I did find some motivation for me!  I found several different (but similar) proofs of the formulas:

sin(x+y) = sin x cos y + cos x sin y

cos(x + y) = cos x cos y - sin y sin x

Can I tell you a secret?  I never had any idea where these formulas came from.  As a math teacher, I'm embarrassed to say that, but it is true!

So, I decided to give this proof a try during class last Monday.  It went about as I expected - some students appreciated the proof...some students' eyes started to glaze over.  But, I felt like I had done my duty and showed them some of the the beauty of mathematics.  Formulas that we use come from unlikely places - someone had to think of them...they didn't just appear out of thin air.

Here are links to a few products that I have that encourage students to see the "proof" in mathematics.  What proofs do you show your class?




Friday, January 20, 2017

Using a Desmos Card Sort in High School Math - Ambiguous Case



We are a 1:1 ipad school and I try to incorporate the iPad into class whenever it works.

I have used Desmos for graphing and to make a Christmas graphing project (see post here: Using Desmos to Graph).

But, I just recently noticed a new feature [new to me at least!] on the teacher side of Desmos.

You can make your own Card Sort!  I am currently teaching the Ambiguous Case in PreCalculus.  Ugh....seriously complicated and sometimes difficult for students to understand.  Plus...I was getting observed during this lesson :)